Heegaard splittings and homeotopy groups of small Seifert fibre spaces
This is the detailed version of the authors' announcement which appeared in C. R. Acad. Sci., Paris, Sér. I 303, 19-22 (1986; Zbl 0596.57010). It completes the result that homotopy implies isotopy for diffeomorphisms of compact Seifert fiber spaces. In fact, in the present paper this is proved for Seifert fiber spaces \(M\) fibering over the 2-sphere \(S^ 2\) with 3 exceptional fibers of orders \((2,3,p)\), \(p>5\), and \((3,3,q)\), \(q>2\) (including the Poincaré homology 3-sphere of type (2,3,5)). These are exactly the Seifert fiber spaces for which the theorem had not been known before, so it finishes the Seifert case and leaves the hyperbolic non-Haken case as the important remaining challenge. The proof depends on another main result of the paper, a certain characterization of Heegaard splittings of minimal genus 2 of Seifert fiber spaces over \(S^ 2\) with 3 exceptional fibers (i.e., with some exception, each Heegaard splitting is vertical). This then served as one cornerstone in the subsequent complete classification, up to isotopy, of Heegaard splittings of genus 2 of such Seifert fiber spaces. In particular, each Seifert fiber space \(M\) as in the beginning has a unique Heegaard splitting, up to isotopy (except for certain Brieskorn varieties which have 2), so one can assume that it is preserved by a given diffeomorphism which can also be assumed to commute with the canonical involution coming from the Heegaard splitting of genus 2. This exhibits \(M\) as a 2-fold covering of the 3- sphere \(S^ 3\) branched over a Montesinos link, so the situation can be projected to \(S^ 3\), and results about symmetries of Montesinos links can be used to show that every diffeomorphism is isotopic to a fiber- preserving one. For these, it is easy to see that homotopy implies isotopy.
isotopic to a fiber-preserving diffeomorphism
diffeomorphisms of compact Seifert fiber spaces
Seifert fiber spaces over \(S^ 2\) with 3 exceptional fibers
scientific article; zbMATH DE number 59795
scientific article